ch4 effective interest

4-*Lecture slides to accompanyEngineering Economy7th edition

Leland BlankAnthony TarquinChapter 4Nominal and Effective Interest Rates 2012 by McGraw-Hill All Rights Reserved

2012 by McGraw-Hill All Rights Reserved

4-*LEARNING OUTCOMESUnderstand interest rate statementsUse formula for effective interest ratesDetermine interest rate for any time periodDetermine payment period (PP) and compounding period (CP) for equivalence calculationsMake calculations for single cash flowsMake calculations for series and gradient cash flows with PP CPPerform equivalence calculations when PP < CPUse interest rate formula for continuous compoundingMake calculations for varying interest rates 2012 by McGraw-Hill All Rights Reserved


4-*Interest Rate StatementsThe terms nominal and effective enter into consideration when the interest period is less than one year.Interest period (t) period of time over which interest is expressed. For example, 1% per month.New time-based definitions to understand and rememberCompounding period (CP) Shortest time unit over which interest is charged or earned. For example,10% per year compounded monthly.Compounding frequency (m) Number of times compounding occurs within the interest period t. For example, at i = 10% per year, compounded monthly, interest would be compounded 12 times during the one year interest period. 2012 by McGraw-Hill All Rights Reserved


4-*Understanding Interest Rate Terminology A nominal interest rate (r) is obtained by multiplying an interest rate that is expressed over a short time period by the number of compounding periods in a longer time period: That is:

r = interest rate per period x number of compounding periodsExample: If i = 1% per month, nominal rate per year is r = (1)(12) = 12% per year)IMPORTANT: Nominal interest rates are essentially simple interest rates. Therefore, they can never be used in interest formulas.

Effective rates must always be used hereafter in all interest formulas. Effective interest rates (i) take compounding into account (effective rates can be obtained from nominal rates via a formula to be discussed later). 2012 by McGraw-Hill All Rights Reserved


2012 by McGraw-Hill All Rights Reserved4-*More About Interest Rate TerminologyThere are 3 general ways to express interest rates as shown belowWhen no compounding period is given, rate is effectiveWhen compounding period is givenand it is not the same as interest period, it is nominalWhen compounding period is givenand rate is specified as effective, rate is effective over stated period


Effective Annual Interest Rates4-* 2012 by McGraw-Hill All Rights ReservedNominal rates are converted into effective annual rates via the equation:ia = (1 + i)m 1

where ia = effective annual interest rate i = effective rate for one compounding period m = number times interest is compounded per yearExample: For a nominal interest rate of 12% per year, determine the nominal and effective rates per year for (a) quarterly, and (b) monthly compoundingSolution:(a) Nominal r / year = 12% per yearEffective i / year = (1 + 0.03)4 1 = 12.55% per year(b) Nominal r /month = 12/12 = 1.0% per yearEffective i / year = (1 + 0.01)12 1 = 12.68% per year Nominal r / quarter = 12/4 = 3.0% per quarter


4-*Effective Interest RatesNominal rates can be converted into effective rates for any time period via the following equation:i = (1 + r / m)m 1 where i = effective interest rate for any time period r = nominal rate for same time period as i m = no. times interest is compd in period specified for iExample: For an interest rate of 1.2% per month, determine the nominaland effective rates (a) per quarter, and (b) per year(a) Nominal r / quarter = (1.2)(3) = 3.6% per quarterEffective i / quarter = (1 + 0.036/3)3 1 = 3.64% per quarter(b) Nominal i /year = (1.2)(12) = 14.4% per yearEffective i / year = (1 + 0.144 / 12)12 1 = 15.39% per yearSolution:Spreadsheet function is = EFFECT(r%,m) where r = nominal rate per period specified for i 2012 by McGraw-Hill All Rights Reserved


4-*Equivalence Relations: PP and CPNew definition: Payment Period (PP) Length of time between cash flows In the diagram below, the compounding period (CP) is semiannual and the payment period (PP) is monthly $10000 1 2 3 4 5 F = ?A = $8000i = 10% per year, compounded quarterly 0 1 2 3 4 5 6 7 8 YearsSemi-annual periodsSimilarly, for the diagram below, the CP is quarterly and the payment period (PP) is semiannual Semi-annual PP 2012 by McGraw-Hill All Rights Reserved


4-*Single Amounts with PP > CPThere are two equally correct ways to determine i and n(1) The i must be an effective interest rate, and(2) The time units on n must be the same as those of i (i.e., if i is a rate per quarter, then n is the number of quarters between P and F)Method 1: Determine effective interest rate over the compounding period CP, and set n equal to the number of compounding periods between P and FMethod 2: Determine the effective interest rate for any time period t, and set n equal to the total number of those same time periods. 2012 by McGraw-Hill All Rights Reserved


4-*Example: Single Amounts with PP CPFor monthly rate, 1% is effective [n = (5 years)(12 CP per year = 60] F = 10,000(F/P,1%,60) = $18,167(b) For a quarterly rate, effective i/quarter = (1 + 0.03/3)3 1 = 3.03%F = 10,000(F/P,3.03%,20) = $18,167(c) For an annual rate, effective i/year = (1 + 0.12/12)12 1 = 12.683%F = 10,000(F/P,12.683%,5) = $18,167effective i per monthmonthseffective i per quarterquarterseffective i per yearyearsi and n must alwayshave same time unitsi and n must alwayshave same time unitsi and n must alwayshave same time units 2012 by McGraw-Hill All Rights Reserved


Series with PP CP4-*For series cash flows, first step is to determine relationship between PP and CPFirst, find effective i per PP Example: if PP is in quarters, must find effective i/quarter(2) Second, determine n, the number of A values involved Example: quarterly payments for 6 years yields n = 46 = 24Determine if PP CP, or if PP < CP 2012 by McGraw-Hill All Rights ReservedNote: Procedure when PP < CP is discussed later


4-*Example: Series with PP CPSolution:First, find relationship between PP and CPPP = six months, CP = one month; Therefore, PP > CPSince PP > CP, find effective i per PP of six monthsStep 1. i /6 months = (1 + 0.06/6)6 1 = 6.15%Next, determine n (number of 6-month periods)Step 2: n = 10(2) = 20 six month periodsFinally, set up equation and solve for F F = 500(F/A,6.15%,20) = $18,692 (by factor or spreadsheet) How much money will be accumulated in 10 years from a deposit of $500 every 6 months if the interest rate is 1% per month? 2012 by McGraw-Hill All Rights Reserved


4-*Series with PP < CPTwo policies: (1) interperiod cash flows earn no interest (most common) (2) interperiod cash flows earn compound interestFor policy (1), positive cash flows are moved to beginning of the interest period in which they occur and negative cash flows are moved to the end of the interest periodNote: The condition of PP < CP with no interperiod interest is the only situation in which the actual cash flow diagram is changedFor policy (2), cash flows are not moved and equivalent P, F, and A values are determined using the effective interest rate per payment period 2012 by McGraw-Hill All Rights Reserved


4-*Example: Series with PP < CPSolution:A person deposits $100 per month into a savings account for 2 years. If $75 is withdrawn in months 5, 7 and 8 (in addition to the deposits), construct the cash flow diagram to determine how much will be in the account after 2 years at i = 6% per year, compounded quarterly. Assume there is no interperiod interest. Since PP < CP with no interperiod interest, the cash flow diagram must be changed using quarters as the time periods0 1 2 3 4 5 6 7 8 9 10 21 24300300300300300MonthsQuarters1 2 3 7 875150F = ? tothis 0 1 2 3 4 5 6 7 8 9 10 23 2410075fromF = ?this7575 2012 by McGraw-Hill All Rights Reserved


4-*Continuous CompoundingTake limit as m to find the effective interest rate equation

i = er 1Example: If a person deposits $500 into an account every 3 months at an interest rate of 6% per year, compounded continuously, how much will be in the account at the end of 5 years? Solution: Payment Period: PP = 3 months Nominal rate per three months: r = 6%/4 = 1.50% Effective rate per 3 months: i = e0.015 1 = 1.51%F = 500(F/A,1.51%,20) = $11,573 2012 by McGraw-Hill All Rights Reserved


4-*Varying RatesFor the cash flow shown below, find the future worth (in year 7) at i = 10% per year.Solution:When interest rates vary over time, use the interest rates associated with their respective time periods to find P Example: Find the present worth of $2500 deposits in years 1 through 8 if the interest rate is 7% per year for the first five years and 10% per year thereafter. P = 2,500(P/A,7%,5) + 2,500(P/A,10%,3)(P/F,7%,5) = $14,68314,683 = A(P/A,7%,5) + A(P/A,10%,3)(P/F,7%,5)A = $2500 per year 2012 by McGraw-Hill All Rights Reserved


i = (1 + r / m)m 14-*Summary of Important PointsMust understand: interest period, compounding period, compounding frequency, and payment period Always use effective rates in interest formulasInterest rates are stated different ways; must know how to get effective ratesFor single amounts, make sure units on i and n are the same 2012 by McGraw-Hill All Rights Reserved


4-*Important Points (contd)For uniform series with PP < CP and no interperiod interest, move cash flows to match compounding periodFor continuous compounding, use i = er 1 to get effective rateFor the cash flow shown below, find the future worth (in year 7) at i = 10% per year.For varying rates, use stated i values for respective time periods 2012 by McGraw-Hill All Rights ReservedFor uniform series with PP CP, find effective i over PP


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ch4 effective interest

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